Large diamond cuboctahedron

3D view of a truncated cuboctahedron ( animation )

The large rhombic cuboctahedron (also called truncated cuboctahedron ) is a polyhedron ( polyhedron ) that is one of the Archimedean solids . It is made up of 12  squares , 8  hexagons and 6  octagons . A square, a hexagon and an octagon each form a corner of the room.

The body that is dual to the truncated cuboctahedron is the hexakisoctahedron .

Formulas

Sizes of a truncated cuboctahedron with edge length a
volume	${\displaystyle V=2\,a^{3}\left(11+7{\sqrt {2}}\right)}$
Surface area	${\displaystyle A_{O}=12\,a^{2}\left(2+{\sqrt {2}}+{\sqrt {3}}\right)}$
Umkugelradius	${\displaystyle R={\frac {a}{2}}{\sqrt {13+6{\sqrt {2}}}}}$
Edge ball radius	${\displaystyle r={\frac {a}{2}}{\sqrt {12+6{\sqrt {2}}}}}$
Diagonal angle  ( octagon - hexagon )  ≈ 125 ° 15 ′ 52 ″	${\displaystyle \cos \,\alpha _{1}=-{\frac {1}{3}}{\sqrt {3}}}$
Diagonal angle (octagon - square )  = 135 °	${\displaystyle \cos \,\alpha _{2}=-{\frac {1}{2}}{\sqrt {2}}}$
Diagonal angle  (hexagon square)  ≈ 144 ° 44 ′ 8 ″	${\displaystyle \cos \,\alpha _{3}=-{\sqrt {\frac {2}{3}}}}$
Corners solid angle  = 1.25 π	${\displaystyle \Omega =2\pi -\arccos \left(-{\frac {1}{2}}{\sqrt {2}}\right)}$

Web links

Commons : Truncated Cuboctahedron  - collection of images, videos and audio files

Wiktionary: truncated cuboctahedron  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Large rhombic cuboctahedron . In: MathWorld (English).